Two functions with C^n continuity over an interval:Which one's the smoothest?

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July 6th 2012, 01:13 PM
x3bnm

Two functions with C^n continuity over an interval:Which one's the smoothest?

According to the definition of Smooth function - Wikipedia, the free encyclopedia:

Functions that have derivatives of all orders are called smooth.

Suppose I've two different functions $f(x) = 2x^2$ and $g(x) = 4x^2$ of continuity $C^n$ over an interval $[0,4]$ . Because both of these functions has derivatives of all order these two functions are smooth.

What I like to know is which one of these two are smoothest?

Is it possible to kindly help me detect which function( $f(x)$ or $g(x)$ ) is the smoothest over the given interval?
July 7th 2012, 01:16 PM
HallsofIvy

Re: Two functions with C^n continuity over an interval:Which one's the smoothest?

How are you defining "smoothest"? Some times texts will say that if f is differentable "n" times and g is differentiable "m" times, with n> m, then f is "smoother" than g. But you are using "smooth" to mean what those texts would call "infinitely smooth" so that does not apply here.
July 8th 2012, 02:30 PM
x3bnm

Re: Two functions with C^n continuity over an interval:Which one's the smoothest?

Quote:

Originally Posted by HallsofIvy

How are you defining "smoothest"? Some times texts will say that if f is differentable "n" times and g is differentiable "m" times, with n> m, then f is "smoother" than g. But you are using "smooth" to mean what those texts would call "infinitely smooth" so that does not apply here.

Thanks HallsofIvy for answering my question. Yes I meant $f(x) = x^3$ is smoother than $g(x) = x^2$ because $x^3$ is one time more differentiable than $x^2$ . I found the answer. Originally my problem was like this:

Suppose I've a acceleration vs. time graph where $x$ means time in millisecond and $y$ means acceleration in $\text{inch/ms}^2$

I've 2 sets of data with 4 points in each set. I've to find the approximated curve in each set that is smoother than the other(means one is more differentiable than other).

I found the solution by the way. All I have to do is curve fit in each set using Linear Algebra. Thanks for helping me see the way.